Planetary Gear Set Calculator

Planetary (Epicyclic) Gear Set A planetary gear set consists of: Sun gear (S): center gear, driven by input in many configurations Planet gears (P): mesh with both sun and ring; mounted on carrier Ring gear (R): outer internal gear; surrounds the planets Carrier (C): holds the planet gear axes; can be output or held fixed

Fundamental Equation (Willis Equation) (ω_R − ω_C) / (ω_S − ω_C) = −N_S / N_R Where ω = angular velocity (RPM), N = number of teeth. This single equation governs all possible configurations.

Three Common Configurations

1. Fixed Ring (R locked): ω_R = 0 Gear ratio = ω_S / ω_C = 1 + N_R / N_S Output (carrier) is slower than input (sun). Used in: first gear of auto transmissions.

2. Fixed Sun (S locked): ω_S = 0 Gear ratio = ω_R / ω_C = 1 + N_S / N_R Output (carrier) is slower than input (ring). Used in: reverse or overdrive in some transmissions.

3. Fixed Carrier (C locked): ω_C = 0 Gear ratio = ω_S / ω_R = −N_R / N_S (negative = reverse rotation) Output (ring) turns SLOWER than input (sun), and in the opposite direction. Since a ring gear always has more teeth than the sun it encloses, N_R/N_S is always greater than 1, so this configuration is always a reduction. What makes it distinctive is the direction reversal, not the ratio. It is how a gearbox gets reverse without a separate idler.

Gear Constraints Planet teeth N_P: determined by geometry. N_R = N_S + 2 × N_P (mesh condition, internal gear) Assembly condition: (N_R + N_S) / number of planets = integer

Applications Automatic transmissions (multiple planetary sets in series for different gear ratios). Bicycle hub gears (Sturmey-Archer, Shimano Nexus). Wind turbine pitch drives. Helicopter gearboxes. Clock and watch escapements (historical use).